Are there ways to solve equations with multiple variables?

Multiplying by the common denominator $7!$, we get

$$ 2520 a_2 + 840 a_3 + 210 a_4 + 42 a_5 + 7 a_6 + a_7 = 3600$$ Take this mod $7$:

$$ a_7 \equiv 2 \mod 7$$ Since $0 \le a_7 < 7$, this means $a_7 = 2$, and then (substituting this and dividing by $7$): $$ 360 a_2 + 120 a_3 + 30 a_4 + 6 a_5 + a_6 = 514$$ Taking this mod $6$: $$ a_6 \equiv 4 \mod 6$$ So $a_6 = 4$.
Continuing the process, we get $$ a_5 = 0,\ a_4 = 1,\ a_3 = 1,\ a_2 = 1 $$

Well, if the variables can take only integer values, then you have a Diophantine equation. The field of math would be number theory.

There is no general method to solve a Diophantine equation, only for special cases (e.g. yours, which happens to be linear).

When you cannot solve a diophantine equation

The MRDP theorem shows that there is no systematic method to determine whether any given diophantine equation has a solution. So if by "solve" you include figuring out whether there is no solution, then it is absolutely impossible in general. Even if you are guaranteed that it has finitely many solutions, you still cannot systematically find all of them. The reason is that otherwise we can use such an oracle $D$ to solve the halting problem as follows.

Let $H$ be the following program on input $(P,x)$:

  Output the following program $Q$ with input $n$:

    If $n = 0$ then:

      Run $P(x)$.

      Accept.

    Reject.

Then clearly $H(P,x)$ is a program that accepts a finite RE (recursively-enumerable) set. By MRDP we can computably convert it to a diophantine equation with finitely many solutions. Now all we have to do is to ask $D$ to find all solutions, and accept $(P,x)$ iff $D$ finds one solution.

Now since $D$ is supposed to always halt, it will find a solution iff $P$ halts on $x$, and hence we have computably solved the halting problem. But that is impossible and therefore $D$ cannot exist.

In summary, even if you guarantee that a diophantine equation has at most a single solution, there is in general no systematic procedure to find if there is a solution, not to say find the solution!

When you can solve a diophantine equation

For a specific diophantine equation, one may be able to find and prove the solutions via necessarily ad-hoc methods. The above shows that we cannot do better than ad-hoc methods, although each ad-hoc method may solve a large class of diophantine equations. For example, congruence mod $n$ for some fixed $n$ can easily show the non-existence of solutions for a large class of equations. It turns out that there are equations for which there are no solutions but such that congruences will never be able to rule out all possibilities.

Another situation in which we can solve a diophantine equation is if we are told that there are at least $k$ solutions, and required to find only $k$ of them. In this case we can iterate through all possibilities until we have found $k$ solutions.

Finally of course there is the trivial case where we are told to find all solutions within some finite set.